The Kerr Metric, General Relativity, and Rotating Black Holes

The Schwarzschild, Kerr, Reissner–Nordström, and Kerr-Newmann metrics: an introduction

The curvature of space-time around a black hole (Source)

Strictly speaking, black holes do not exist. Actually, holes, of any kind, do not exist. You can talk about holes, of course. “There’s a hole in the wall”. Sure, you can say that. You can also detail the hole however you’d like. But I am sure that you do not think that there is a “thing” made out of nothingness in the wall. No. To talk about the hole is actually a way of talking about the wall. What really exists is the wall. The wall is made out of bricks, atoms, protons, leptons, whatever. To say that there is a hole in the wall is just to say that the wall has a certain topology — a topology such that not every closed curve on the surface of the wall can be contracted to a single point. The hole is not a thing. The hole is a property of the wall.

So, black holes. What are black holes a property of? Spacetime? Great, but what is spacetime? And what about spacetime are we even concerned with? Spacetime is the set of all events of all things. Everything that has happened, happens, and will happen, is just an element of spacetime. Spacetime is not a thing, it is just the way we represent the relations among all things.

For decades, black holes have headlined the thought experiments that physicists seek refuge in. These invisible spheres form when matter becomes so concentrated that everything within a certain distance gets trapped by its gravity. A black hole, in that sense, just represents an incredibly dense region of space that curves spacetime to such an extreme that nothing, not even light, is allowed to escape.

To make math infinitely more convenient, physicists tend to generalize black holes to death. We assume a spherical event horizon that doesn’t have any rotational motion of its own. Though, to be honest, that’s far from reality. Black holes aren’t just static spheres of matter existing ad infinitum.

On that note, if you haven’t already read my previous article on the Bekenstein-Hawking entropy, I’d recommend you do. Because then, it becomes abundantly clear why we’re even concerned about the difference between stationary and spinning black holes.

Black Holes: a Perspective on Disorder

The mathematical description of a rotating black hole is given by the Kerr metric:

The metric in Boyer-Lindquist coordinates

Now, admittedly, this looks like a nightmare to work with but we don’t need to necessarily worry about it yet. What I’d like for you to take away from this, however, is that the Kerr metric is a solution to Einstein’s field equations:

More maths but again, this isn’t what we’ll be focusing on. The reason I’m even including this is so that you understand the motivation behind what follows. In fact, after bringing this up, we can think of general relativity the following way.

Based on our theoretical and experimental triumphs, we understand that Einstein's field equations seem to be a great description of our Universe. Or at least, the best description mankind has seen so far. Different solutions to these equations dictate different scenarios. For instance, we can use the Minkowski metric to describe empty flat space and we’ve got the Schwarzschild metric to describe the behavior of spacetime around a stationary black hole.

The Minkowski metric — a description of empty space

The Schwarzschild metric — a description of stationary black holes

Needless to say, finding solutions to Einstein’s field equations is hard. But we have found some and all evidence suggests that solutions to Einstein’s field equations are not merely theoretical. Each of them manifests in a different astrophysical phenomenon.

The first relevant thing is that for a rotating black hole, the event horizon is not the only important boundary we must consider. The event horizon is the limit beyond which the escape velocity becomes greater than the speed of light. And yes, this exists for a rotating black hole but it’s not spherical. If our black hole is rotating about the z-axis, it looks something like this:

The boundaries of a Kerr black hole relevant to astrophysics. Note that there are no physical “surfaces” as such. The boundaries are mathematical surfaces, or sets of points in spacetime, relevant to the analysis of the black hole’s properties and interactions. Source.

Now note that we just said the event horizon is not the only boundary we need to consider. Since there’s an outer event horizon, there must be an inner one too. And there is. But it’s important to know that this is just the result of the mathematics that the Kerr metric leaves us with. The Kerr metric is a description of spacetime around spinning black holes. The mathematics postulates the existence of an inner horizon. Though this is not necessarily a “physical” surface — just a mathematical one.

The outer event horizon is the natural result of the Kerr mathematics. The inner one? Not so much. It’s just there to make the math make sense.

I’m being a little weird about this and there’s a reason for it. The inner event horizon is only hand-wavy because the Kerr metric is intended to describe the behavior of spacetime around spinning black holes. That’s why it’s not necessarily the best description of what’s inside the black hole.

Secondly, as far as we know, there’s actually no way for us to find out whether the inner event horizon exists or not. Because not even light can escape the outer event horizon, we don’t really know what goes on inside. And for these reasons, we won’t delve too deep into the inner horizon. Seemingly, it does exist. At least that’s what the mathematics dictates.

But we’re not done yet. Another boundary worth mentioning is the outer stationary limit surface. Beyond this boundary, an effect known as “frame dragging” hits you. All matter beyond the stationary limit surface must rotate in the same direction as the black hole. Nothing, not even light, can move against the direction in which the black hole rotates.

In essence, if a photon is moving opposite to the spin of the black hole, then once it’s within the stationary limit surface, it must do a 180 of sorts and align itself with the black hole’s rotation. As it crosses the surface, it must move in the same direction as the black hole’s rotation.

This is what we mean by frame dragging. If we were to find a particular reference frame to look at a particle inside the stationary limit surface, then in order for us to see the particle as stationary, our reference frame must move with the rotation of the black hole. The black hole’s rotation is dragging our frame.

This effect is quite literally the manifestation of warped spacetime. Since it’s rotating, we get something that looks like this:

1: Frame-dragging of a rotating black hole, as viewed from above. [ResearchGate]

As the black hole rotates faster, the distance between the event horizon and the limit surface increases. This actually allows two distinct boundaries to be observed. The region between the limit surface and the outer event horizon is what we call the ergosphere. This is a region where particles are subjected to such extreme warping of spacetime that they’re forced to rotate in the same direction as the black hole. Crucially, however, the ergosphere still represents the region of space where a particle can escape the gravitational pull of a black hole. It would be at relativistically high speeds but in principle, it’s possible.

This is partly why energy can be extracted by a black hole. Theorized by Sir Roger Penrose, the Penrose process is a means by which particles in the ergosphere, can actually extract energy from a rotating black hole. This certainly deserves an article of its own but it boils down to the idea that particles can carry energy away from a black hole. The maximum amount of energy gain possible for a single particle via this process is 20.7% of its mass in the case of an uncharged black hole (assuming the best case of maximal rotation of the black hole). The energy is taken from the rotation of the black hole, so there is a limit on how much energy one can extract through the Penrose process.

The Singularity

The Kerr metric’s theoretical successes don’t just stop there. It also extends itself to describing the singularity at the heart of the black hole. Some of you might be familiar with the idea of a singularity but for those of you that aren’t, a singularity is a region of infinite density. Density is mass over volume and as the volume tends towards zero, i.e., becomes incredibly small, density grows to infinity. At these singularities, our mathematics breaks and our theories blow up. Unfortunately, however, we may never really know what happens at the center of a black hole because information never escapes the event horizon. Still, it’s worth thinking about.

For stationary black holes, the math tells us that the singularity must be a point. An infinitesimally small region of space with infinitely high mass. For a rotating black hole, on the other hand, we understand that the singularity must take the form of a ring. A very thin, very narrow ring around the axis of rotation. But here’s the thing. The Kerr metric doesn’t just describe black holes. For an object to be a black hole, it must have an event horizon. The Kerr metric tells us that the faster a black hole spins, the smaller its event horizon is. Eventually, the event horizon doesn’t exist anymore. You just have what’s called the naked singularity.

Technically, this tells us that if a naked singularity exists, we can directly verify our theoretical progress. Till now, no such singularity has been observed. In fact, Roger Penrose even hypothesized our inability to ever observe a naked singularity. It’s called the cosmic censorship hypothesis. The hypothesis states that, whenever a body collapses so completely as to result in the formation of a singularity, a black hole will be formed so that the singularity will be hidden behind the horizon, and thus completely unobservable to anyone outside the black hole. But Penrose didn’t just stop there. No. He went a step beyond. In 1969, he posited that no naked singularities exist in the Universe. If his hypothesis is indeed true then we’re left to our last set of theoretical defenses against understanding what really happens inside a black hole.

Stephen Hawking and Roger Penrose (Source: University of Oxford)

In this article and the previous one, we’ve discussed two types of black holes: stationary and rotating. Given by the Schwarzschild and Kerr metrics respectively, these black holes do exist and we have all reason to believe that our mathematical description is not off by a long shot. But of course, nature doesn’t stop there. What we’ve spoken about so far has included the unsaid assumption that our black holes are neutral.

Charge is a conserved quantity. So, if a body with an electric charge falls into a black hole then the charge must be conserved. Consequently, the black hole gains it. We can now invoke two more types of black holes: stationary-charged ones and rotating-charged ones. The metrics for these are also solutions to Einstein’s field equations:

The Reissner–Nordström metric is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M.

The Kerr–Newman metric gives the analogous solution for a charged, rotating body. The Kerr–Newman metric is the most general stationary solution of the Einstein-Maxwell equations in that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass. It generalizes the Kerr metric by taking into account the field energy of an electromagnetic field, in addition to describing rotation. It is one of a large number of various different electrovacuum solutions, that is, solutions to the Einstein-Maxwell equations which account for the field energy of an electromagnetic field. Such solutions do not include any electric charges other than that associated with the gravitational field and are thus termed vacuum solutions.

Anyway, I suppose we’ll save the math for another day. I hope that was a sufficient introduction to the intuition behind it all. As always, thank you for reading!